ECF III: Self-Consistency Gradient Flow Dynamics of Self-Constrained Systems

We present the rigorous derivation of the dynamical equations governing the convergence of self-constrained systems to criticality, as the metric unfolding of the Emergence-Convergence Framework (ECF). Starting from the strength dimension of the unique ontological postulate that existence is correlation, we construct the positive-definite kernel through metricization of the weak order structure, define the self-consistency functional S(λ), and prove that the evolution of self-constrained systems must follow the negative gradient flow of S(λ). The resulting λ evolution equation takes the continuous form dλ/dτ = −S'(λ). We rigorously prove that λ = 1/2 is the unique and globally asymptotically stable fixed point, with stability guaranteed by the Lyapunov function V(λ) = (λ−1/2)². The robustness of this conclusion is established: any configuration hypothesis satisfying the qualitative constraints of monotonicity, boundary divergence, and self-dual symmetry yields λ=1/2 as the unique minimum of S(λ). We establish the conceptual bridge between the gradient flow dynamics and the emergence of physical time. Time does not emerge through a unique quantitative mapping from the logical ordinal τ, but through a convergence into a permissible interval bounded from below by the critical slowing-down scale and from above by the global correlation scale of the system. This interval constitutes the range of viable coarse-grainings within which continuous macroscopic time is a valid effective description. The qualitative framework for time emergence is closed at the C-tier; the precise quantitative boundaries of the permissible interval are marked as open problems (D-tier), to be addressed by the gravitational module of the framework. These equations do not replace any known physical law. Instead, they provide a unified dynamical foundation for the origin and selection of physical laws themselves, fundamentally resolving the fine-tuning problem of cosmic criticality. They also supply the metric-face inputs required by the conditional construction theorem for continuous geometry in the companion paper on mathematics (ECF II).